I don't have any personal experience with Google Hosting, but most hosts that give you an email address will also give you all of the server information that you need to check and send email using a regular email program, like Outlook or ThunderBird.

If Google's Hosting does too, then you can check and send email from within Gmail by going to Settings | Accounts & Import, scroll down to "Check mail using POP3" and click the "Add POP3 Email Account" button. In there you just enter the POP3 (Incoming) server information, the SMTP (Outgoing) server information, and some other basic info, and you'll be able to use Gmail as your regular email client. When you want to send an email from snazzy@mywebsite.com you'll be able to choose it from a dropdown combobox in the Compose area when sending a new email.

It'll also autocheck for email on it's own, but one thing to note, Gmail is very slow when autochecking outside POP3 accounts. It only checks about once an hour. I personally like mine to autocheck about once every 5-10 minutes, and Gmail has no way to change this. So as long as you don't mind the length of time between server checks, it's good. You can manually check whenever you want by going to the same place and clicking the "Check now" link though.